Geodesic finite elements for Cosserat rods

We introduce geodesic finite elements as a new way to discretize the non-linear configuration space of a geometrically exact Cosserat rod. These geodesic finite elements naturally generalize standard one-dimensional finite elements to spaces of functions with values in a Riemannian manifold. For the special orthogonal group, our approach reproduces the interpolation formulas of Crisfield and Jelenic. Geodesic finite elements are conforming and lead to objective and path-independent problem formulations. We introduce geodesic finite elements for general Riemannian manifolds, discuss the relationship between geodesic finite elements and coefficient vectors, and estimate the interpolation error. Then we use them to find static equilibria of hyperelastic Cosserat rods. Using the Riemannian trust-region algorithm of Absil et al. we show numerically that the discretization error depends optimally on the mesh size. Copyright © 2009 John Wiley & Sons, Ltd.

[1]  Peter Deuflhard,et al.  Asymptotic Mesh Independence of Newton's Method Revisited , 2004, SIAM J. Numer. Anal..

[2]  Arne Marthinsen,et al.  Interpolation in Lie Groups , 1999, SIAM J. Numer. Anal..

[3]  S. Antman.  Nonlinear problems of elasticity , 1994 .

[4]  J. Spillmann,et al.  CoRdE: Cosserat rod elements for the dynamic simulation of one-dimensional elastic objects , 2007, SCA '07.

[5]  Nicholas I. M. Gould,et al.  Trust Region Methods , 2000, MOS-SIAM Series on Optimization.

[6]  J. C. Simo,et al.  A three-dimensional finite-strain rod model. Part II: Computational aspects , 1986 .

[7]  Werner Wagner,et al.  THEORY AND NUMERICS OF THREE-DIMENSIONAL BEAMS WITH ELASTOPLASTIC MATERIAL BEHAVIOUR ∗ , 2000 .

[8]  Robert E. Mahony,et al.  Optimization Algorithms on Matrix Manifolds , 2007 .

[9]  M. Crisfield,et al.  Objectivity of strain measures in the geometrically exact three-dimensional beam theory and its finite-element implementation , 1999, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[10]  M. Spivak.  A comprehensive introduction to differential geometry , 1979 .

[11]  F. Béthuel,et al.  The approximation problem for Sobolev maps between two manifolds , 1991 .

[12]  R. Kornhuber.  Adaptive monotone multigrid methods for nonlinear variational problems , 1997 .

[13]  I. Holopainen.  Riemannian Geometry , 1927, Nature.

[14]  Oliver Sander,et al.  Multidimensional coupling in a human knee model , 2008 .

[15]  Ahmed K. Noor,et al.  Mixed models and reduced/selective integration displacement models for nonlinear analysis of curved beams , 1981 .

[16]  J. Nash.  The imbedding problem for Riemannian manifolds , 1956 .

[17]  T. Seidman,et al.  Equilibrium states of an elastic conducting rod in a magnetic field , 1988 .

[18]  Robert L. Pego,et al.  Hamiltonian dynamics of an elastica and the stability of solitary waves , 1996 .

[19]  Carlo Sansour,et al.  Multiplicative updating of the rotation tensor in the finite element analysis of rods and shells – a path independent approach , 2003 .